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Let $(X, d)$ be a compact metric space and let $mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I colon mathcal{M}(X) to R$ by $I(mu) = int_X int_X d(x,y) dmu(x) dmu(y)$, and set $M(X) = sup I(mu)$, where $mu$ ranges over the collection of measures in $mathcal{M}(X)$ of total mass 1. The space $(X, d)$ is emph{quasihypermetric} if $I(mu) leq 0$ for all measures $mu$ in $mathcal{M}(X)$ of total mass 0 and is emph{strictly quasihypermetric} if in addition the equality $I(mu) = 0$ holds amongst measures $mu$ of mass 0 only for the zero measure. This paper explores the constant $M(X)$ and other geometric aspects of $X$ in the case when the space $X$ is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are $L^1$-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [Peter Nickolas and Reinhard Wolf, emph{Distance geometry in quasihypermetric spaces. I}, emph{II} and emph{III}].
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Let $(X, d)$ be a compact metric space and let $mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I colon mathcal{M}(X) to R$ by [I(mu) = int_X int_X d(x,y) dmu(x) dmu(y),] and set $M(X) = sup I(mu)$, where $mu$ ranges over the collection of signed measures in $mathcal{M}(X)$ of total mass 1. The metric space $(X, d)$ is quasihypermetric if for all $n in N$, all $alpha_1, ..., alpha_n in R$ satisfying $sum_{i=1}^n alpha_i = 0$ and all $x_1, ..., x_n in X$, one has $sum_{i,j=1}^n alpha_i alpha_j d(x_i, x_j) leq 0$. Without the quasihypermetric property $M(X)$ is infinite, while with the property a natural semi-inner product structure becomes available on $mathcal{M}_0(X)$, the subspace of $mathcal{M}(X)$ of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of $(X, d)$, the semi-inner product space structure of $mathcal{M}_0(X)$ and the Banach space $C(X)$ of continuous real-valued functions on $X$; conditions equivalent to the quasihypermetric property; the topological properties of $mathcal{M}_0(X)$ with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-$*$ topology and the measure-norm topology on $mathcal{M}_0(X)$; and the functional-analytic properties of $mathcal{M}_0(X)$ as a semi-inner product space, including the question of its completeness. A later paper [Peter Nickolas and Reinhard Wolf, Distance Geometry in Quasihypermetric Spaces. II] will apply the work of this paper to a detailed analysis of the constant $M(X)$.
Let $(X, d)$ be a compact metric space and let $mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I colon mathcal{M}(X) to R$ by [ I(mu) = int_X int_X d(x,y) dmu(x) dmu(y), ] and set $M(X) = sup I(mu)$, where $mu$ ranges over the collection of signed measures in $mathcal{M}(X)$ of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $mathcal{M}(X)$ when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum defining $M(X)$, sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the finiteness of $M(X)$.
Let $(X, d)$ be a compact metric space and let $mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I colon mathcal{M}(X) to R$ by [ I(mu) = int_X int_X d(x,y) dmu(x) dmu(y), ] and set $M(X) = sup I(mu)$, where $mu$ ranges over the collection of signed measures in $mathcal{M}(X)$ of total mass 1. This paper, with two earlier papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and II], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $mathcal{M}(X)$ when equipped with a natural semi-inner product. Specifically, this paper explores links between the properties of $M(X)$ and metric embeddings of $X$, and the properties of $M(X)$ when $X$ is a finite metric space.
A metric space $X$ is rigid if the isometry group of $X$ is trivial. The finite ultrametric spaces $X$ with $|X| geq 2$ are not rigid since for every such $X$ there is a self-isometry having exactly $|X|-2$ fixed points. Using the representing trees we characterize the finite ultrametric spaces $X$ for which every self-isometry has at least $|X|-2$ fixed points. Some other extremal properties of such spaces and related graph theoretical characterizations are also obtained.
Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitati
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